Filter circuit

ABSTRACT

Disclosed is a filter circuit with an order of three or more, comprising at least one means for amplifying an in-band signal, wherein the frequency response of the filter output has a desirable attenuation characteristic obtainable with the order of the filter circuit. The gain of the amplifying means is variably controlled.

FIELD OF THE INVENTION

The present invention relates to a filter circuit and particularly to anon-chip filter circuit suitably integrated on an LSI chip.

BACKGROUND OF THE INVENTION

FIG. 1 is a diagram showing the configuration of a conventional filtercircuit. The conventional filter circuit comprises two filters, whichhave respectively desired attenuation characteristic, and which aredisposed in front of and behind an amplifier, respectively, to realize asteep frequency response and a high voltage gain between an input and anoutput of the filter circuit. More specifically, the conventional filtercircuit, as shown in FIG. 1, comprises a 5th-order Low Pass Filter(abbreviated to LPF) and a 4th-order LPF disposed in the preceding andsucceeding stages of the amplifier, respectively. In an article(Non-Patent Document 1), there is disclosed a receiver in which two LPFsare disposed in front of and behind an amplifier. These two LPFs, whichare termed channel selection filters, for which an attenuationcharacteristic of 80 dB is required at the bandwidth between neighboringchannels. Since a maximum voltage gain of approximately 80 dB isdemanded in the baseband unit in such a direct conversion receiver,amplifiers are provided both in front of the preceding stage 5th-orderLPF and behind the succeeding stage 4th-order LPF.

These three amplifiers are variable gain amplifiers (abbreviated to‘VGA’). A high gain amplifier having a gain of 80 dB not only an inputnoise, but also a noise generated in the amplifier, resulting in highnoise level at the output and poor signal-to-noise ratio (S/N ratio).This is why LPFs are provided separately at different stages.

Each of LPFs described in the article (Non-Patent Document 1) is anactive filter and generates a noise inside the circuit, and hence theplacement of noise sources and filters becomes important in order toachieve a desired S/N ratio.

In case where the amplifier in FIG. 1 is assumed to be a linear typeone, the relationship between the input signal and the output signal isexpressed by the product of the transfer function of the 5th-order LPFin the preceding stage of the amplifier, the voltage gain of theamplifier, and the transfer function of the 4th-order LPF in thesucceeding stage of the amplifier.

Accordingly, in the frequency response of the entire circuit shown inFIG. 1, the −3 dB cut-off frequency (f_(c)) of the entire circuit islower than the −3 dB cut-off frequency (f_(c)+f₁) (where f₁>0) of the5th-order LPF in the preceding stage of the amplifier and the −3 dBcut-off frequency (f_(c)+f₂) (where f₂>0) of the 4th-order LPF in thesucceeding stage of the amplifier.

The attenuation characteristic of the 5th-order LPF in the precedingstage of the amplifier has a slope of −30 dB/Octave as from the −3 dBcut-off frequency (f_(c)+f₁) in case of the 5th-order LPF having theButterworth characteristic. The attenuation characteristic of the4th-order LPF in the succeeding stage of the amplifier has a slope of−24 db/Octave as from the −3 dB cut-off frequency (f_(c)+f₂) in case ofthe 4th-order LPF having the Butterworth characteristic.

On the other hand, since the attenuation characteristic of a 9th-orderLPF has a slope of −54 dB/Octave as from the −3 dB cut-off frequency ifthe 9th-order LPF is a Butterworth filter, an attenuation amount of 54dB might be obtained at a frequency which is twice the −3 dB cut-offfrequency (2f_(c)). However, only an attenuation amount of less than 50dB is obtained at the frequency 2f_(c) in FIG. 1.

In order to realize an attenuation amount of 50 dB or more at thefrequency 2f_(c), it is necessary to:

-   -   increase the order of a filter by one, changing the 4th-order        LPF in the succeeding stage of the amplifier to a 5th-order LPF,        or    -   change the 5th-order LPF in the preceding stage of the amplifier        to a 6th-order LPF.

[Non-Patent Document 1]

C. Shi et al., “Design of A Low-Power CMOS Baseband Circuit for WidebandCDMA Testbed,” Proceedings of International Symposium on Low PowerElectronics and Design (ISLPED 2000), 2000 Jul. 26-27, pp. 222-224, FIG.1.

SUMMARY OF THE DISCLOSURE

The conventional technology described above has the following problems.

The first problem is that it is difficult for the filter circuit toachieve a desired attenuation characteristic corresponding to the orderthereof. The reason is that the −3 dB cut-off frequency of the filtercircuit composed by a plurality of filters connected to each other in acascade connection, becomes lower than the −3 dB cut-off frequency ofeach of the plurality of filters constituting the circuit.

The second problem is that in order to realize a desired attenuationcharacteristic, the number of circuit components and a chip size areincreased. The reason is that the orders of filters have to be madehigher to realize a desired attenuation characteristic.

The third problem is an increase in current consumption of the filtercircuit. It is because the increase in the order of the filters incursthe increase in the number of circuit elements.

The present invention has been invented to solve the above-mentionedproblems.

The invention disclosed in the present application is configured asfollows:

A filter circuit in accordance with one aspect of the present invention,which has an order of three or more, comprises at least one means foramplifying an in-band signal, such that the frequency response at anoutput of said filter circuit has a desirable attenuation characteristicobtainable with the order of the filter circuit. Preferably, in thepresent invention, the gain of the amplifying means is variablycontrolled.

The meritorious effects of the present invention are summarized asfollows. The present invention enables a desirable attenuationcharacteristic to be achieved while keeping the order of the filter to aminimum. The reason is that in the present invention the filter circuitcomprises an amplifier for making the frequency response of the filter adesirable attenuation characteristic.

According the present invention, the current consumption can be reduced.The reason is that in the present invention, the order of the filter iskept to a minimum.

According the present invention, the chip area can be reduced. Thereason is that in the present invention, the filter including theamplifier is realized while keeping the order of the filter to a minimumin the present invention.

According the present invention, an excellent noise characteristic canbe achieved. The reason is that in the present invention, the noiseamplified by the amplifier can be attenuated by a filter in a succeedingstage in the present invention.

Still other features and advantages of the present invention will becomereadily apparent to those skilled in this art from the followingdetailed description in conjunction with the accompanying drawingswherein only the preferred embodiments of the invention are shown anddescribed, simply by way of illustration of the best mode contemplatedof carrying out this invention. As will be realized, the invention iscapable of other and different embodiments, and its several details arecapable of modifications in various obvious respects, all withoutdeparting from the invention. Accordingly, the drawing and descriptionare to be regarded as illustrative in nature, and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing a conventional filter circuit.

FIG. 2 is a diagram showing the filter circuit of the present invention.

FIG. 3 is a diagram illustrating a circuit realized by gm-C filters of a3rd-order inverse Chebychev LPF with an inserted voltage amplifierhaving a voltage gain of Gv of the present invention.

FIG. 4 is a diagram showing the circuit of a 3rd-order LCR ladder LPFfor explaining the present invention.

FIG. 5 is a drawing illustrating the characteristic of the 3rd-orderinverse Chebychev LPF for explaining the present invention.

FIG. 6 is a drawing illustrating the normalized amplitude characteristicof the 3rd-order inverse Chebychev LPF for explaining the presentinvention.

FIG. 7 is a drawing illustrating the normalized phase characteristic ofthe 3rd-order inverse Chebychev LPF for explaining the presentinvention.

FIG. 8 is a drawing illustrating the simulation values of the amplitudecharacteristic of the 3rd-order inverse Chebychev LPF of the presentinvention.

FIG. 9 is a drawing illustrating the normalized amplitude characteristicof a 5th-order inverse Chebychev LPF for explaining the presentinvention.

FIG. 10 is a drawing illustrating the normalized phase characteristic ofthe 5th-order inverse Chebychev LPF for explaining the presentinvention.

FIG. 11 is a diagram illustrating a circuit realized by gm-C filters ofthe 5th-order inverse Chebychev LPF with two inserted voltage amplifiershaving voltage gains of Gv1 and Gv2, respectively, of the presentinvention.

FIG. 12 is a drawing illustrating the simulation values of the amplitudecharacteristic of the 5th-order inverse Chebychev LPF of the presentinvention.

FIG. 13 is a drawing illustrating the simulation values of the amplitudecharacteristic of another 5th-order inverse Chebychev LPF of the presentinvention.

FIG. 14 is a drawing illustrating the simulation values of the amplitudecharacteristic of the 5th-order inverse Chebychev LPF of the presentinvention. (G_(V1)=G_(V2)=10)

FIG. 15 is a diagram illustrating a circuit realized by gm-C filters ofthe 5th-order inverse Chebychev LPF showing another embodiment of thepresent invention.

FIG. 16 is a drawing illustrating the simulation values of the amplitudecharacteristic of the 5th-order inverse Chebychev LPF showing anotherembodiment of the present invention.

FIG. 17 is a drawing illustrating the simulation values of the amplitudecharacteristic of the 5th-order inverse Chebychev LPF showing anotherembodiment of the present invention. (G_(V)=10)

FIG. 18 is a diagram showing the circuit of a 5th-order LCR ladder LPFfor explaining the present invention.

FIG. 19 is a drawing illustrating the normalized amplitudecharacteristic of the 5th-order inverse Chebychev LPF for explaining thepresent invention.

FIG. 20 is a drawing illustrating the normalized phase characteristic ofthe 5th-order inverse Chebychev LPF for explaining the presentinvention.

FIG. 21 is a diagram illustrating a circuit realized by a gm-C filter ofa 2nd-order LPF for explaining the present invention.

FIG. 22 is a diagram illustrating a filter circuit showing anotherembodiment of the present invention.

FIG. 23 is a diagram illustrating a circuit realized by gm-C filters ofa 5th-order LPF showing another embodiment of the present invention.

PREFERRED EMBODIMENTS OF THE INVENTION

Embodiments for carrying out the present invention will be described indetail with reference to the drawings.

FIG. 2 is a drawing showing the configuration of a mode of carrying outthe present invention. Referring to FIG. 2, in the present mode, afifth-order filter is divided into a 1st-order filter block and2nd-order filter blocks, and amplifiers are inserted between neighboringfilter blocks. Each 2nd-order filter block is not a complete filter suchas a 2nd-order Butterworth filter or 2nd-order Chebychev filter. Thecombination of all 2nd-order filter blocks realizes a completen^(th)-order Butterworth filter or n^(th)-order Chebychev filter.

The design of such a 1st-order filter block and 2nd-order filter blocksis able to be carried out by using for example a filter circuitrealization method called the Biquad method. Note that, in the Biquadmethod, a transfer function of a desired filter characteristic isdecomposed into a product of 1st-order or 2nd-order transfer function. A1st-order filter circuit and a 2nd-order filter circuit are realized bya 1 st-order transfer function and a 2nd-order transfer function,respectively.

The 2nd-order filter circuit, based on the Biquad method, can benormally realized by an RC active filter circuit consisting of op-amps(operational-amplifiers), resistors (Rs), and capacitors (Cs), or a gm-Cfilter consisting of OTAs (Operational Transconductance Amplifiers) andcapacitors (Cs), but it cannot necessarily be realized only by passiveelements such as inductors (Ls), resistors (Rs), and capacitors (Cs).

Note that, since an op-amp has a voltage input and a voltage output, therelationship between the input and the output is defined as a voltagegain. Since an OTA has a voltage input and a current output, therelationship between the input and the output is defined as atransconductance which is expressed as gm. Further, in a gm-C filter, acapacitor C is utilized as an equivalent inductance L according to thegyrator theory.

As an embodiment of the present invention, a specific example in which a3rd-order inverse Chebychev filter is realized will be described first.

FIG. 3 is a diagram illustrating the embodiment of the present inventionadapted to a 3rd-order inverse Chebychev filter. An inverse ChebychevLPF has a maximally flat passband compared to a Butterworth LPF. Havingtransmission zeros in the stopband, the inverse Chebychev LPF has anequi-ripple, and has its attenuation characteristic limited. However,the inverse Chebychev LPF is used as a channel selection filter in areceiver because a steep attenuation characteristic is able to beobtained, and is suited for data transmission because it has an in-bandphase characteristic superior than that of a Butterworth LPF.

However, since there has been no appropriate text that describes thedesign technique of inverse Chebychev filters, not many designers in thefield are familiar with them. For reference, the design technique of a3rd-order inverse Chebychev LPF will be explained in detail.

[The General Concept of the Design Approach]

First, an inverse transformation procedure will be described. In the“inverse Chebychev filter”, there is provided an adjective “inverse”.This “inverse” corresponds to the inverse of an inverse function. We arefamiliar as inverse functions with an exponential function and alogarithmic function, and a square and a √{square root over ( )} (squareroot). The relationship between a voltage and a current in a bipolartransistor or a MOS transistor is such an inverse function. However, afunction or an inverse function is not always used clearly with adiscrimination of the difference between them.

It is necessary to discriminate a Chebychev filter from an inverseChebychev filter clearly. Here, “inverse” means that the relationshipbetween the passband and the stopband is functionally inversed.

Assuming that the transfer function of a Chebychev filter is T_(C)(s),the following is derived in a first step:|T _(I C)(jω)|²=1−|T _(C)(jω)|²  (1)and the following is derived in a second step:|H(jω)|² =|T _(I C)(j/ω)|²  (2)

In a third step, the transfer function H(s) of the inverse Chebychevfilter is derived from the equation (2).

The equation (1) corresponds to the operation of replacing the passbandwith the stopband, and the equation (2) corresponds to the operation ofexchanging the frequency axes.

In other words, the transformation from LPF to HPF (or from HPF to LPF)is carried out by the replacement of the passband with the stopband, andthe restoration from HPF to LPF (or from LPF to HPF) is carried out bythe exchange of the frequency axes from ω to 1/ω.

As a result, the equiripple characteristic in the passband moves to thestopband, and the characteristic of monotonous attenuation along with afrequency ω in the stopband changes to the flat characteristic of thepassband.

A specific example will be described. First, a 3rd-order Chebychev LPFwill be described in the below.

The transfer function TC(s) of a 3rd-order Chebychev LC ladder LPF shownin FIG. 4 is given as follows, if R1=R3=1, C1=C2=c, and L2=1:

$\begin{matrix}{{T_{c}(s)} = {\frac{2}{\left( {{cs} + 1} \right)\left( {{cls}^{2} + {ls} + 2} \right)} = \frac{2}{{c^{2}{ls}^{3}} + {2{cls}^{2}} + {\left( {{2c} + l} \right)s} + 2}}} & (3)\end{matrix}$Note that the numerator is set to 2 so that T_(C)(0)=1.

The square of the amplitude characteristic is given:

$\begin{matrix}{{{T_{c}({j\omega})}}^{2} = \frac{4}{\left( {2 - {2{cl}\;\omega^{2}}} \right)^{2} + \left\{ {{\left( {{2c} + l} \right)\omega} - {c^{2}l\;\omega^{3}}} \right\}^{2}}} & (4)\end{matrix}$

When the passband is replaced with the stopband, the following is given:

$\begin{matrix}{{1 - {{T_{c}({j\omega})}}^{2}} = \frac{\left\{ {{\left( {{2c} - 1} \right)\omega} - {c^{2}l\;\omega^{3}}} \right\}^{2}}{\left( {2 - {2{cl}\;\omega^{2}}} \right)^{2} + \left\{ {{\left( {{2c} + l} \right)\omega} - {c^{2}l\;\omega^{3}}} \right\}^{2}}} & (5)\end{matrix}$

Further, when the frequency axes are exchanged, the following is given:

$\begin{matrix}{{1 - {{T_{c}\left( {j/\omega} \right)}}^{2}} = \frac{\left\{ {{\left( {{2c} - l} \right)\omega} - {c^{2}l}}\; \right\}^{2}}{\left( {{2\omega^{3}} - {2{cl}\;\omega}} \right)^{2} + \left\{ {{\left( {{2c} + l} \right)\omega^{2}} - c^{2}} \right\}^{2}}} & (6)\end{matrix}$Note the relationship between the numerator and the second term of thedenominator.

The second term of the denominator in the equation (6):{(2c+l)ω²−c1²l}²and the numerator:{(2c−l)ω²−c1²l}²They are settled by:{(2c∓l)ω²−c1²l}²Looking at each coefficient of ω^(n), since the numerator has − and thedenominator has +, the coefficient of ω^(n) in the numerator is smallerthan the coefficient of ω^(n) in the denominator.

Accordingly, the transfer function H(s) of the 3rd-order Chebychev LPFis given as follows:

$\begin{matrix}{{H(s)} = {\frac{{\left( {{2c} - l} \right)s^{2}} + {c^{2}l}}{{2s^{3}} + {\left( {{2c} + l} \right)s^{2}} + {2{cls}} + {c^{2}l}} = \frac{{\left( {{2c} - l} \right)s^{2}} + {c^{2}l}}{\left( {s + c} \right)\left( {{2s^{2}} + {ls} + {cl}} \right)}}} & (7)\end{matrix}$

The equation (3) may also be applied to a 3rd-order Butterworth LPF.Further, if √{square root over ( )} (the square root) is applied in the(6), the amplitude characteristic of the 3rd-order inverse Chebychev LPFindicated by the equation (7) can be obtained.

From the equation (7), the denominator is realized by a 1st-order LPFand a 2nd-order LPF, while the numerator is realized by an ellipticcapacitor.

Therefore, the 3rd-order inverse Chebychev LPF is realized by using a3rd-order elliptic gm-C filter, and the number of OTAs (OperationalTransconductance Amplifier) necessary is determined by the order of thefilter as before.

In order to derive the transfer function H(s) of the inverse Chebychevfilter from the transfer function T_(c)(s) of a Chebychev filter, thecoefficients are exchanged in order in the denominator e.g. exchangingthe highest-order coefficient and the lowest-order coefficient(constant).

The numerator is realized by adding a term as²+b that realizes anelliptic characteristic.

APPLICATION EXAMPLE

As an example, the constants of the normalized 0.01 dB ripple 3rd-orderChebychev LPF are substituted such that R1=R3=1 Ω, C1=C3=c=1.1811F, andL2=I=1.8214H. The graph of the normalized amplitude value in thefrequency response of the 3rd-order Chebychev LPF is shown in FIG. 5.

(0.01 dB ripple 3rd-order Chebychev LPF:

$\left. {{T_{c}(s)} = \frac{2}{{2.540848\mspace{11mu} s^{3}} + {4.302511\mspace{11mu} s^{2}} + {4.1836\mspace{11mu} s} + 2}} \right)$

Further, the passband is replaced by the stopband (1−|T(jω)|²), and thefrequency axes are exchanged (1−|T(j/ω)|²). Finally, when the squareroot (√{square root over ( )}) is applied, the amplitude characteristicof the 3rd-order inverse Chebychev LPF is obtained.

(3rd-order inverse Chebychev LPF:

$\left. {{H(s)} = \frac{{0.540692454\mspace{11mu} s} + 2.540848}{{2s^{3}} + {4.1836\mspace{11mu} s^{2}} + {4.302511\mspace{20mu} s} + 2.540848}} \right)$

This process can be traced back in FIG. 5. The amplitude characteristic(a) of the 0.01 dB ripple 3rd-order Chebychev LPF and the amplitudecharacteristic (c) of a 3rd-order Butterworth LPF approximately overlap,however, the equiripple characteristic in the stopband does notattenuate up to 30 dB in the amplitude characteristic (e) of the inverseChebychev LPF. The in-band equiripple characteristic of the ChebychevLPF is 3 dB at maximum.

Therefore, the equiripple characteristic in the stopband of the inverseChebychev LPF is less than or equal to −10.666 dB, and a largeattenuation amount with an equiripple characteristic in the stopband canbe obtained while maintaining an excellent flat characteristic in thepassband.

ANOTHER APPLICATION EXAMPLE

Further, the transfer function of a 3rd-order inverse Chebychev LPFwhose equiripple characteristic in the stopband is improved to 40 dB isderived in case of n=3 (3rd order), as follows:

$\begin{matrix}{a = {\frac{1}{n}{\sinh^{- 1}\left( \frac{1}{ɛ} \right)}}} & (8)\end{matrix}$whereε=√{square root over (10^(α) ^(max) ^(/10)−1)}  (9)where α_(max)(<3 dB) in √{square root over ( )} indicates the equiripplevalue.sin h ⁻¹(χ)=ln(χ+√{square root over (χ²+1)})  (10)cos h ⁻¹(χ)=ln(χ+√{square root over (χ²−1)})  (11)

The root is derived as follows:−α_(K)=sin h(a)·sin(Φ_(K))  (12)±β_(K)=cos h(a)·cos(Φ_(K))  (13)

where:

$\begin{matrix}{{\phi_{k} = {{\frac{{2k} + 1}{n}\frac{\pi}{2}\mspace{31mu} k} = 0}},1,\ldots\mspace{11mu},{{2n} - 1}} & (14)\end{matrix}$

In order to obtain an equiripple attenuation characteristic of −40 dBwith an inverse Chebychev LPF, αmax=0.0004342 dB and ε=0.01. From theequation (1), a=1.766142155 and sin h(a)=2.83862838728775. Further,ΦK=30°, 90°, 150°, and the following is given:p ₁ ,p ₂=−1.41931419364388±j2.60640717096099p ₃=−2.83862838728775

The transfer function of a 0.0004342 dB ripple 3rd-order Chebychev LPFis given as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{25.0021}{\left( {s + 2.83862839} \right)\left( {s^{2} + {2.83862839\mspace{11mu} s} + 8.80781112} \right)}} & (15)\end{matrix}$

The denominator in the expression (15) is expanded as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{25.0021}{s^{3} + {5.677257\mspace{14mu} 14.42726\mspace{11mu} s^{2}} + {16.86562\mspace{11mu} s} + 25.0021}} & (16)\end{matrix}$

Note that the cut-off frequency ω_(hp)(hp:half-power) is given:

$\begin{matrix}{\omega_{h\; p} = {{\cosh\left( {\frac{1}{n}{\cosh^{- 1}\left( \frac{1}{e} \right)}} \right\}}❘}} & (17)\end{matrix}$

In the case of the 0.0004342 dB ripple 3rd-order Chebychev LPF, thecut-off frequency ω_(hp)=3.00957237 holds. When the cut-off frequencyω_(hp)=1, the transfer function of the 0.0004342 dB ripple 3rd-orderChebychev LPF is given as the follows:

$\begin{matrix}{{T_{c}(s)} = \frac{0.917196}{s^{3} + {1.8864\mspace{11mu} s^{2}} + {1.862056\mspace{11mu} s} + 0.917196}} & (18)\end{matrix}$

When the denominator in the expression (18) is factorized, the transferfunction is given as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{0.917196}{\left( {s + 0.9432} \right)\mspace{14mu}\left( {s^{2} + {0.9432\mspace{11mu} s} + 0.97243} \right)}} & (19)\end{matrix}$

Comparing the coefficients of the equations (3) and (18), c=1.060220526and l=1.939882512 hold.

When these values are substituted in the equation (7), the transferfunction H(s) of the 3rd-order inverse Chebychev LPF is given asfollows:

$\begin{matrix}\begin{matrix}{{H(s)} = \frac{{0.180559\mspace{11mu} s^{2}} + 2.180559}{{2s^{3}} + {4.060324\mspace{11mu} s^{2}} + {4.113407\mspace{11mu} s} + 2.180559}} \\{= \frac{{0.090279\mspace{11mu} s^{2}} + 1.09028}{s^{3} + {2.030162\mspace{11mu} s^{2}} + {2.056703\mspace{11mu} s} + 1.09028}}\end{matrix} & (20)\end{matrix}$

The factorization of the denominator in the expression (20) gives:

$\begin{matrix}\begin{matrix}{{H(s)}\; = \;\frac{{0.180559\mspace{11mu} s^{2}}\; + \; 2.180559}{\left( {s + 1.060221} \right)\left( {{2\mspace{11mu} s^{2}} + {1.939883\mspace{11mu} s} + 2.056703} \right)}} \\{= \frac{{0.090279\mspace{11mu} s^{2}}\; + \; 1.09028}{\left( {s + 1.060221} \right)\left( {s^{2} + {0.969941\mspace{11mu} s} + 1.028352} \right)}}\end{matrix} & (21)\end{matrix}$

In FIG. 6, the amplitude characteristic (c) of the 3rd-order inverseChebychev LPF is shown along with that (a) of the original 0.0004342 dBripple 3rd-order Chebychev LPF and that (b) of the 3rd-order ButterworthLPF. While the transmission zero of the 3rd-order inverse Chebychev LPFis ω=2.17 (attenuation band ripple height: −26.4 dB) in case of theripple characteristic of the original 3rd-order Chebychev LPF shown inFIG. 5 being 0.01 dB, the transmission zero of the 3rd-order inverseChebychev LPF is ω=3.47(attenuation band ripple height: −40.0 dB) incase of the ripple characteristic of the original 3rd-order ChebychevLPF being 0.0004342 dB.

Further, the phase characteristic (c) of the 3rd-order inverse ChebychevLPF is shown in FIG. 7 along with that (a) of the original 0.0004342 dBripple 3rd-order Chebychev LPF and that (b) of the 3rd-order ButterworthLPF. The phase characteristic of the 3rd-order inverse Chebychev LPF isapproximately the same as the phase characteristic of the original0.0004342 dB ripple 3rd-order Chebychev LPF and the phase characteristicof the 3rd-order Butterworth LPF.

DESIGN EXAMPLE

Now we finally return to the case of the 3rd-order Chebychev gm-C LPFshown in FIG. 3. Because a single-end OTA has rather a high secondarydistortion and an unsatisfactory signal amplitude, the gm-C LPF in FIG.3 is fully differential. The gm-C filter, which comprises OTAs andcapacitors, performs an integrating operation by charging the capacitorwith a current which is proportionate to an input voltage to realize afilter characteristic. In the example shown in FIG. 3, a 1st-order gm-CLPF comprises an OTA 1 which differentially receives a voltage Vin, andan OTA2 which has differential inputs connected to the connection pointsbetween differential outputs of the OTA1 and one ends of two capacitorsC₁ whose other ends are grounded, and has differential outputs connectedto the other ends of the two capacitors C1 respectively. A 2nd-ordergm-C LPF comprises an OTA 3 which has differential inputs connected tothe connection points between differential outputs of an amplifier (gainGv) 11 and one ends of two capacitors C_(L2) whose other ends aregrounded, and has differential outputs connected to the other ends ofthe two capacitors C_(L2), respectively, an OTA 4 which has differentialinputs connected to the other ends of the capacitors C_(L2), and an OTA5 which has differential inputs connected to the connection pointsbetween differential outputs of the OTA4 and one ends of two capacitorsC₃ whose other ends are grounded, and has differential outputs connectedto the other ends of the two capacitors C₃, respectively. Thedifferential outputs of the OTA 5 are cross-connected to thedifferential inputs of the OTA 4. The 2nd-order gm-C LPF furthercomprises an OTA 6 which has differential inputs connected to the otherends of two capacitors C₃ and has differential outputs connected to theother ends of the two capacitors C₃. The differential outputs of the OTA4 are feedback-connected to the connection points between thedifferential inputs of the OTA 3 and the differential outputs of theamplifier via the capacitors C₂. The mutual conductances of the OTA1through the OTA 6 are set to gm₁ through gm₆.

In the preceding stage of the amplifiers of the 3rd-order inverseChebychev gm-C LPF shown in FIG. 3, there is provided a 1st-order gm-CLPF whose transfer function H₁(s) is given as follows:

$\begin{matrix}{{H_{1}(s)} = {\frac{V_{out}}{V_{i\; n}} = \frac{\frac{2g_{m\; 1}}{C_{1}}}{s + \frac{2g_{m\; 2}}{C_{1}}}}} & (22)\end{matrix}$

In the succeeding stage of the amplifiers of the 3rd-order inverseChebychev gm-C LPF shown in FIG. 3, there is provided a 2nd-orderelliptic gm-C LPF whose transfer function H₂(s) is given as follows:

$\begin{matrix}{{H_{2}(s)} = {\frac{V_{out}}{V_{i\; n}} = \frac{{\frac{C_{2}}{C_{2} + C_{3}}s^{2}} + \frac{4g_{m\; 3}g_{m\; 4}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}{s^{2} + {s\frac{2g_{m\; 6}}{C_{2} + C_{3}}} + \frac{4g_{m\; 4}g_{m\; 6}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}} & (23)\end{matrix}$

Therefore, the transfer function defined by the product of H₁(s) andH₂(s) is given as follows:

$\begin{matrix}\begin{matrix}{{H(s)} = {{{H_{1}(s)} \cdot {H_{2}(s)}} = {\frac{\frac{2g_{m\; 1}}{C_{1}}}{2 + \frac{2g_{m\; 2}}{C_{1}}} \cdot \frac{{\frac{C_{2}}{C_{2} + C_{3}}s^{2}} + \frac{4g_{m\; 3}g_{m\; 4}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}{s^{2} + {s\frac{2g_{m\; 6}}{C_{2} + C_{3}}} + \frac{4g_{m\; 4}g_{m\; 6}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}}} \\{= \frac{{\frac{2g_{m\; 1}C_{2}}{C_{1}\left( {C_{2} + C_{3}} \right)}s^{2}} + \frac{8g_{m\; 1}g_{m\; 3}g_{m\; 4}}{C_{1}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}{\begin{matrix}{s^{2} + {\left( {\frac{2g_{m\; 2}}{C_{1}} + \frac{2g_{m\; 6}}{C_{2} + C_{3}}} \right)s^{2}} + \left\{ {\frac{4g_{m\; 2}g_{m\; 6}}{C_{1}\left( {C_{2} + C_{3}} \right)} +} \right.} \\{{\left. \frac{4g_{m\; 4}g_{m\; 6}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)} \right\} s} + \frac{8g_{m\; 2}g_{m\; 4}g_{m\; 6}}{C_{1}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}\end{matrix}}}\end{matrix} & (24)\end{matrix}$

When g_(m1)=g_(m2)=g_(m3)=g_(m4)=g_(m5)=g_(m6)=g_(m), the followingequation holds:

$\begin{matrix}\begin{matrix}{{H(s)} = \frac{{\frac{2g_{m}C_{2}}{C_{1}\left( {C_{2} + C_{3}} \right)}s^{2}} + \frac{8g_{m}^{3}}{C_{1}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}{\begin{matrix}{s^{3} + {\left( {\frac{2g_{m}}{C_{1}} + \frac{2g_{m}}{C_{2} + C_{3}}} \right)s^{2}} + \left\{ {\frac{4g_{m}^{2}}{C_{1}\left( {C_{2} + C_{3}} \right)} +} \right.} \\{{\left. \frac{4g_{m}^{2}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)} \right\} s} + \frac{8g_{m}^{3}}{C_{1}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}}\end{matrix}}} \\{= \frac{{\frac{C_{2}C_{L\; 2}}{4g_{m}^{2}}s^{2}} + 1}{\begin{matrix}{{\frac{C_{1}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}}{8g_{m}^{3}}s^{3}} + \left\{ {\frac{C_{1}C_{L\; 2}}{4g_{m}^{2}} +} \right.} \\{{\left. \frac{C_{L\; 2}\left( {C_{2} + C_{3}} \right)}{4g_{m}^{2}} \right\} s^{2}} + {\left( {\frac{C_{1}}{2g_{m}} + \frac{C_{L\; 2}}{2g_{m}}} \right)s} + 1}\end{matrix}}}\end{matrix} & (25)\end{matrix}$

Here, the coefficients are compared with the ones in the equation (21)and the constants are determined.

The transfer function H(s) of the 3rd-order inverse Chebychev LPF withan attenuation band of −40 dB is given as follows with the denominatorand numerator factorized:

$\begin{matrix}{{H(s)} = \frac{{0.090279\mspace{11mu} s^{2}} + 1.09028}{\left( {s + 1.060221} \right)\left( {s^{2} + {0.969941\mspace{11mu} s} + 1.028352} \right)}} & (26)\end{matrix}$

This transfer function is decomposed into a 1st-order LPF:

$\begin{matrix}{{{H_{1}(s)} = \frac{1.060221}{s + 1.060221}}{{and}\mspace{14mu} a\mspace{14mu} 2{nd}\text{-}{order}\mspace{14mu}{elliptic}\mspace{14mu}{LPF}\text{:}}} & (27) \\{{H_{2}(s)} = \frac{{0.08515111\mspace{11mu} s^{2}} + 1.02835164}{s^{2} + {0.969941\mspace{11mu} s} + 1.028352}} & (28)\end{matrix}$

When the coefficients of the equations (22) and (27) are compared:

$\begin{matrix}{\frac{2g_{m\; 1}}{C_{1}} = {\frac{2g_{m\; 2}}{C_{1}} = 1.060221}} & (29)\end{matrix}$Therefore, g_(m1)=g_(m2).

This stands to reason because the numerator in the equation (27) is setto 1.060221, equal to the zero-order coefficient of the denominator. Inother words, H₁(0)=1 holds.

And H₂(0)=1 holds. Further, the following equations hold:

$\begin{matrix}{\frac{2g_{m\; 6}}{C_{2} + C_{3}} = 0.969941} & (30) \\{\frac{4g_{m\; 4}g_{m\; 5}}{C_{s\; 2}\left( {C_{2} + C_{3}} \right)} = 1.028352} & (31) \\{\frac{C_{2}}{C_{2} + C_{3}} = 0.08515111} & (32) \\{\frac{4g_{m\; 3}g_{m\; 4}}{C_{s\; 2}\left( {C_{2} + C_{3}} \right)} = 1.028352} & (33)\end{matrix}$

Here, when g_(m3)=g_(m4)=g_(m5)=g_(m6)=5 μS, and f_(c)=1.920 MHz, thefollowing equation holds from the equation (29):

$\begin{matrix}{C_{1} = {\frac{10\mu}{1.060221 \times 2\pi \times 1.920\mspace{11mu} M} = {0.7818483\mspace{14mu}\lbrack{pF}\rbrack}}} & (34)\end{matrix}$

The equation (31) is equal to the equation (33), and when the equation(30) is substituted into them:

$\begin{matrix}{C_{L\; 2} = {\frac{10\mu}{\left( {1.028352/0.969941} \right) \times 2\pi \times 1.920\mspace{14mu} M} = {0.781848169\mspace{14mu}\lbrack{pF}\rbrack}}} & (35)\end{matrix}$

From the equation (30):

$\begin{matrix}{{C_{2} + C_{3}} = {\frac{10\mu}{0.969941 \times 2\pi \times 1.920\mspace{14mu} M} = {0.85462105\mspace{14mu}\lbrack{pF}\rbrack}}} & (36)\end{matrix}$

From the equation (32), since C₂=0.093076693C3, the following equationsare given:C₃=0.7818483 [pF]  (37)C₂=0.07277193 [pF]  (38)

In the 3rd-order gm-C LPF shown in FIG. 3, if the OTAs are replaced byvoltage controlled current sources (VCCSs) which are ideal OTAs, usingthe constants calculated above, the 3rd-order gm-C LPF is able to beeasily simulated by SPICE. Here, voltage controlled voltage sources(VCVSs) are inserted between the stages. Note that when VCCSs areconnected as gm₁ and gm₂, the isolation that VCVSs offer cannot beobtained.

Note that all the capacitance values are equal (C₁=C_(L2)=C₃) in thecase of the 3rd-order inverse Chebychev LPF shown here. Further, thedenominator of the equation (28) has a small Q value, Q=1.0455.

In FIG. 8, the frequency response obtained is shown. In FIG. 8, evenwhen the voltage gain GV is varied from 1, the level of cap1 isunchanged, but the levels of cap3 and outp change on a log scale by theamount of the voltage gain GV. In other words, the 3rd-order inverseChebychev LPF is realized by the 1st-order gm-C LPF, the amplifiers, andthe 2nd-order elliptic gm-C LPF.

Embodiment 2

Next, how a 5th-order inverse Chebychev gm-C is designed will bedescribed.

The transfer function H(s) of the 5th-order inverse Chebychev LPF havingan attenuation band of 40 dB is given as follows:

$\begin{matrix}{{H(s)} = \frac{\left( {{0.3463405\mspace{11mu} s^{2}} + 1} \right)\left( {{0.13231949\mspace{11mu} s^{2}} + 1} \right)}{\begin{matrix}{\left( {1 + {0.78555\mspace{11mu} s}} \right)\left( {1 + {0.485496\mspace{14mu} s} + {0.963449465\mspace{11mu} s^{\; 2}}} \right)} \\\left( {1 + {1.271046\mspace{11mu} s} + {0.749386387\mspace{11mu} s^{\; 2}}} \right)\end{matrix}}} & (39)\end{matrix}$

The frequency response of this 5th-order inverse Chebychev LPF having anattenuation band of 40 dB is shown in FIG. 9 (characteristic c in FIG.9), and the phase characteristic is shown in FIG. 10 (characteristic ain FIG. 10).

Here, when H(s) is decomposed into the products of transfer functions of1st-order LPF and 2nd-order LPF respectively:H(s)=H ₁(s)·H ₂(s)·H ₃(s)  (40)H₁(s), H₂(s), and H₃(s) are given as follows:

$\begin{matrix}{{H_{1}(s)} = \frac{1.272993444}{s + 1.272993444}} & (41) \\{{H_{2}(s)} = \frac{{0.35947967\mspace{14mu} 4s^{2}} + 1.037937158}{s^{2} + {0.50914338\mspace{11mu} s} + 1.037937158}} & (42) \\{{H_{3}(s)} = \frac{{0.17657044\mspace{14mu} 7s^{2}} + 1.334425094}{s^{2} + {1.696115678\mspace{14mu} s} + 1.334425094}} & (43)\end{matrix}$

The 5th-order inverse Chebychev gm-C LPF is realized by a 1st-order gm-CLPF and two 2nd-order gm-C LPFs. All g_(m) are equal and when g_(m)=5 μSand f_(C)=1.92 MHz, the capacitance value of each filter is as follows.For the first stage 1st-order gm-C LPF, from:

$\begin{matrix}{{\frac{2g_{m}}{C_{1}} = 1.272993444}{C_{1}\mspace{14mu}{is}\mspace{14mu}{given}\mspace{14mu}{as}\mspace{14mu}{follows}\text{:}}} & (44) \\{C_{1} = {\frac{10\mu}{1.272993444 \times 2\pi \times 1.92\mspace{11mu} M} = {0.6511675\mspace{14mu}\lbrack{pF}\rbrack}}} & (45)\end{matrix}$For the next stage 2nd-order gm-C LPF:

$\begin{matrix}{\frac{2g_{m}}{C_{2} + C_{3}} = 0.503914338} & (46) \\{\frac{C_{2}}{C_{2} + C_{3}} = 0.359479674} & (47) \\{\frac{4g_{m}^{2}}{C_{L\; 2}\left( {C_{2} + C_{3}} \right)} = 1.037937158} & (48)\end{matrix}$

The equation (46) is substituted into the equation (48):

$\begin{matrix}\begin{matrix}{C_{L\; 2} = \frac{10\mu}{\left( {1.037937158/0.503914338} \right) \times 2\pi \times 1.92M}} \\{= {0.4024431676\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (49)\end{matrix}$

From the equation (46):

$\begin{matrix}{{C_{2} + C_{3}} = {\frac{10\mu}{0.503914338 \times 2\pi \times 1.92M} = {1.644985929\mspace{14mu}\lbrack{pF}\rbrack}}} & (50)\end{matrix}$

From the equation (47), since C₂=0.561230704C3, C₂ and C₃ are asfollows:C₂=0.5913390053 [pF]  (51)C₃=1.053646923 [pF]  (52)

Similarly, for the last stage 2nd-order gm-C LPF:

$\begin{matrix}{\frac{2g_{m}}{C_{4} + C_{5}} = 1.696115678} & (53) \\{\frac{C_{4}}{C_{4} + C_{5}} = 0.176570447} & (54) \\{\frac{4g_{m}^{2}}{C_{L\; 4}\left( {C_{4} + C_{5}} \right)} = 1.334425094} & (55)\end{matrix}$

The equation (53) is substituted into the equation (55):

$\begin{matrix}\begin{matrix}{C_{L\; 4} = \frac{10\mu}{\left( {1.334425094/1.696115678} \right) \times 2\pi \times 1.92M}} \\{= {1.053610697\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (56)\end{matrix}$

From the equation (53):

$\begin{matrix}\begin{matrix}{{C_{4} + C_{5}} = \frac{10\mu}{1.696115678 \times 2\pi \times 1.92M}} \\{= {0.4887237386\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (57)\end{matrix}$

Since C₄=0.214432972C5 from the equation (47), C₄ and C₅ are as follows:C₄=0.086294168 [pF]  (58)C₅=0.402429569 [pF]  (59)

FIG. 11 shows the configuration of the 5th-order inverse Chebychev gm-CLPF in which two amplifiers (VCVSs) with voltage gains of G_(V1) andG_(V2) are inserted between the stages of the 1st-order gm-C LPF and thetwo 2nd-order gm-C LPFs. As in the case of the 3rd-order inverseChebychev gm-C LPF, the frequency response is unchanged regardless ofhow the voltage gains G_(V1) and G_(V2) are set. The frequency responsein case of G_(V1)=G_(V2)=1 is shown in FIG. 12. The level (×2.7) at thecap2 terminal is notable.

The transfer functions are exchanged between the two 2nd-order LPFs fromthe equation (39). From

$\begin{matrix}{{H_{2}(s)} = \frac{{0.13733931\mspace{11mu} 5\; s^{2}} + 1.037937158}{s^{2} + {0.503914338\mspace{14mu} s} + 1.037937158}} & (42)^{\prime} \\{{H_{3}(s)} = \frac{{0.46216545\mspace{11mu} 4\mspace{11mu} s^{2}} + 1.334425094}{s^{2} + {1.696115678\mspace{14mu} s} + 1.334425094}} & (43)^{\prime}\end{matrix}$are derived C₁=0.6511675[pF], C_(L2)=0.4024431676[pF],C₂=0.2259212406[pF], C₃=1.419064688[pF], C₄=0.2258712285[pF],C_(L4)=1.053610697[pF], and C₅=0.26285251[pF].

In this case, the frequency response incase of G_(V1)=G_(V2)=1 is shownin FIG. 13, which is identical to FIG. 12.

As evident from the circuit diagram in FIG. 11 and the frequencyresponse graph in FIG. 12, the 5th-order inverse Chebychev gm-C LPF isable to be broken down into three gm-C filter blocks, in which twoamplifiers (VCVSs) of respective voltage gains G_(V1) and G_(V2) areinserted between the 1st-order gm-C LPF and the next stage 2nd-ordergm-C LPF and between the 2nd-order gm-C LPF and another final stage2nd-order gm-C LPF. The placements of these three gm-C LPF blocks aremutually interchangeable.

Further, it is effective to replace an amplifier with an amplifier suchas a VGA, and divide a filter to dispose divided filters in anapplication such as a wireless device. As a concrete example, thefrequency response in case where the voltage gains of the two amplifiers(VCVSs) are set to 10 times (×10) is shown in FIG. 14.

The amplitude of the 5th-order inverse Chebychev gm-C LPF is 100 at thefinal stage output (out9 p), but there is no change in the frequencyresponse.

Embodiment 3

Further, in the case of the 5th-order inverse Chebychev gm-C LPF, the1st-order gm-C LPF and the next stage 2nd-order gm-C LPF are connectedand an amplifier (VCVS) with a voltage gain Gv is inserted between theconnected block and the third stage 2nd-order gm-C LPF as shown in FIG.15. Each constant in this case is derived as follows.

Here, the transfer function of the 5th-order inverse Chebychev gm-C LPFshown in FIG. 15 is given:

$\begin{matrix}{{H(s)} = \frac{2{g_{m\; 1}\left( {{C_{2}C_{L\; 2}s^{2}} + {4g_{m\; 3}g_{m\; 4}}} \right)}\left( {{C_{4}C_{L\; 4}s^{2}} + {4g_{m\; 7}g_{m\; 8}}} \right)}{\begin{matrix}\left\{ {{{C_{L\; 4}\left( {C_{4} + C_{5}} \right)}s^{2}} + {2g_{m\; 10}C_{L\; 4}s} + {4g_{m\; 8}g_{m\; 9}}} \right\} \\\left\{ {{{C_{\;{L\; 2}}\left( {{C_{\; 1}C_{\; 3}} + {C_{\; 1}C_{\; 2}} + {C_{\; 2}C_{\; 3}}} \right)}s^{\; 3}} +} \right. \\{C_{\;{L\; 2}}\left( {{2g_{\;{m\; 6}}C_{\; 1}} + {2g_{\;{m\; 2}}C_{\; 2}} + {2g_{\;{m\; 6}}C_{\; 2}} + {\left. {2g_{\;{m\; 2}}C_{\; 3}} \right)s^{\; 2}} +} \right.} \\\left( {{4g_{\;{m\; 2}}g_{\;{m\; 6}}C_{\;{L\; 2}}} + {4g_{\;{m\; 4}}g_{\;{m\; 5}}C_{\; 1}} +} \right. \\{{\left. {{4g_{\;{m\; 4}}g_{\;{m\; 6}}C_{\; 2}} - {4g_{\;{m\; 3}}g_{\;{m\; 4}}C_{\; 2}}} \right)s} +} \\\left. {8g_{m\; 2}g_{m\; 4}g_{m\; 6}} \right\}\end{matrix}}} & (60)\end{matrix}$

The equation (60) can be decomposed into:

$\begin{matrix}{{{{H_{1}(s)} \cdot {H_{2}(s)}} = \frac{2{g_{m\; 1}\left( {{C_{2}C_{L\; 2}s^{2}} + {4g_{m\; 3}g_{m\; 4}}} \right)}\left( {{C_{4}C_{L\; 4}s^{2}} + {4g_{m\; 7}g_{m\; 8}}} \right)}{\begin{matrix}\begin{matrix}\begin{matrix}{{{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)}s^{3}} +} \\{{{C_{L\; 2}\left( {{2g_{m\; 6}C_{1}} + {2g_{m\; 2}C_{2}} + {2g_{m\; 6}C_{2}} + {2g_{m\; 2}C_{3}}} \right)}s^{2}} +}\end{matrix} \\\left( {{4g_{m\; 2}g_{m\; 6}C_{L\; 2}} + {4g_{m\; 4}g_{m\; 5}C_{1}} + {4g_{m\; 4}g_{m\; 6}C_{2}} -} \right.\end{matrix} \\\left. {{\left. {4g_{m\; 3}g_{m\; 4}C_{2}} \right)s} + {8g_{m\; 2}g_{m\; 4}g_{m\; 5}}} \right\}\end{matrix}}}\mspace{20mu}{and}} & (61) \\{\mspace{20mu}{{H_{3}(s)} = \frac{{C_{4}C_{L\; 4}} + {4g_{m\; 7}g_{m\; 8}}}{{{C_{L\; 4}\left( {C_{4} + C_{5}} \right)}s^{2}} + {2g_{m\; 10}C_{L\; 4}s} + {4g_{m\; 8}g_{m\; 9}}}}} & (62)\end{matrix}$

Here, wheng_(m1)=g_(m2)=g_(m3)=g_(m4)=g_(m5)=g_(m6)=g_(m7)=g_(m8)=g_(m9)=g_(m10)=gm,the following equations hold:

$\begin{matrix}\begin{matrix}{{{H_{1}(s)} \cdot {H_{2}(s)}} = \frac{2{g_{m}\left( {{C_{2}C_{L\; 2}s^{2}} + g_{m}^{2}} \right)}}{\begin{matrix}\begin{matrix}\left\{ {{{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)}s^{3}} +} \right. \\{{2g_{m}{C_{L\; 2}\left( {C_{1} + {2\mspace{14mu} C_{2}} + C_{3}} \right)}s^{2}} +}\end{matrix} \\\left. {{4{g_{m}^{2}\left( {C_{L\; 2} + C_{1}} \right)}s} + {8g_{m}^{3}}} \right\}\end{matrix}}} \\{= \frac{\begin{matrix}{\frac{2\; g_{\; m}\; C_{\; 2}\; s^{\; 2}}{\;{{C_{\; 1}\; C_{\; 3}}\; + \;{C_{\; 1}\; C_{\; 2}}\; + \;{C_{\; 2}\; C_{\; 3}}}} +} \\\frac{8\mspace{11mu} g_{\; m}^{\; 3}}{\mspace{11mu}{C_{\;{L\; 2}}\;\left( {{C_{\; 1}C_{\; 3}} + {C_{\; 1}C_{\; 2}} + {C_{\; 2}C_{\; 3}}} \right)}}\end{matrix}}{\begin{matrix}\begin{matrix}{s^{3} + {\frac{2{g_{m}\left( {C_{1} + {2\mspace{14mu} C_{2}} + C_{3}} \right)}}{{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}}s^{2}} +} \\{{\frac{4{g_{m}^{2}\left( {C_{1} + C_{L\; 2}} \right)}}{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)}s} +}\end{matrix} \\\frac{8g_{m}^{3}}{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)}\end{matrix}}}\end{matrix} & (63) \\\begin{matrix}{{H_{3}(s)} = \frac{{C_{4}C_{L\; 4}} + {4g_{m}^{2}}}{{{C_{L\; 4}\left( {C_{4} + C_{5}} \right)}s^{2}} + {2g_{m}C_{L\; 4}s} + {4g_{m}^{2}}}} \\{= \frac{{\frac{C_{4}}{\left( {C_{4} + C_{5}} \right)}s^{2}} + \frac{4g_{m}^{2}}{C_{L\; 4}\left( {C_{4} + C_{5}} \right)}}{s^{2} + {\frac{2g_{m}}{C_{4} + C_{5}}s} + \frac{4g_{m}^{2}}{C_{L\; 4}\left( {C_{4} + C_{5}} \right)}}}\end{matrix} & (64)\end{matrix}$

Since the equation (61) does not seem to be able to be factorized intothe product of H₁(s) and H₂(s), the equations (41) and (42) areexpanded:

$\begin{matrix}{{{H_{1}(s)} \cdot {H_{2}(s)}} = \frac{{0.457615268\mspace{11mu} s^{2}} + 1.321287197}{\begin{matrix}{s^{3} + {1.776907782\mspace{11mu} s^{2}} + {1.679416807\mspace{11mu} s} +} \\1.32128797\end{matrix}}} & (65)\end{matrix}$

The coefficients of the equations (63) and (65) are identical,therefore:

$\begin{matrix}{\frac{2g_{m}C_{2}}{{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} = 0.457615268} & (66) \\{\frac{8g_{m}^{3}}{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)} = 1.321287197} & (67) \\{\frac{2{g_{m}\left( {C_{1} + {2\mspace{14mu} C_{2}} + C_{3}} \right)}}{{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} = 1.776907782} & (68) \\{\frac{4{g_{m}^{2}\left( {C_{1} + C_{L\; 2}} \right)}}{C_{L\; 2}\left( {{C_{1}C_{3}} + {C_{1}C_{2}} + {C_{2}C_{3}}} \right)} = 1.679416807} & (69)\end{matrix}$

The coefficients of the equations (44) and (64) become identical aswell, therefore:

$\begin{matrix}{\frac{C_{4}}{C_{4} + C_{5}} = 0.176570447} & (70) \\{\frac{4g_{m}^{2}}{C_{L\; 4}\left( {C_{4} + C_{5}} \right)} = 1.334425094} & (71) \\{\frac{2g_{m}}{C_{4} + C_{5}} = 1.696115678} & (72)\end{matrix}$

First, the constant included in H₃(s) are determined:

From the equation (72):

$\begin{matrix}\begin{matrix}{{C_{4} + C_{5}} = \frac{10\mu}{1.696115678 \times 2\pi \times 1.92M}} \\{= {0.4887237383\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (73)\end{matrix}$

Since C₄=0.214432972C5 from the equation (70):C₄=0.086294168 [pF]  (74)C₅=0.402429569 [pG]  (75)

The equation (72) is substituted into the equation (71), and it is givenas follows:

$\begin{matrix}\begin{matrix}{C_{L\; 4} = \frac{10\mu}{\left( {1.334425094/1.696115678} \right) \times 2\pi \times 1.92M}} \\{= {1.053610697\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (76)\end{matrix}$

Next, the constants included in H₁(s)·H₂(s) are determined.

When the equation (67) is divided by the equation (69),

$\begin{matrix}{\frac{2g_{m}}{C_{1} + C_{L\; 2}} = 0.786753587} & (77)\end{matrix}$

From the equation (75), we have:

$\begin{matrix}\begin{matrix}{{C_{1} + C_{L\; 2}} = \frac{10\mu}{0.786753587 \times 2\pi \times 1.92M}} \\{= {1.053610697\mspace{14mu}\lbrack{pF}\rbrack}}\end{matrix} & (78) \\{{\therefore C_{1}} = {{1.053610697\lbrack{pF}\rbrack} - C_{L\; 2}}} & (79)\end{matrix}$

When the equation (68) is divided by the equation (66), we have:

$\begin{matrix}{\frac{C_{1} + {2\mspace{14mu} C_{2}} + C_{3}}{C_{2}} = {{\frac{C_{1}}{C_{2}} + 2 + \frac{C_{3}}{C_{2}}} = 3.882973114}} & (80) \\{{\therefore{\frac{C_{1}}{C_{2}} + \frac{C_{3}}{C_{2}}}} = 1.882973114} & (81)\end{matrix}$

As a result, C₂ is give as follows:

$\begin{matrix}{C_{2} = \frac{C_{1} + C_{3}}{1.882973114}} & (82)\end{matrix}$

When the equations (66) through (69) are filled by using Excel to fitC_(L2) and C₃, the respective constant with which the errors of theequations (66) through (69) are not higher than ±0.02% can be given asfollows:C₁=0.6915107 [pF]  (83)C_(L2)=0.3621 [pF]  (84)C₂=0.6571314 [pF]  (85)C₃=0.54585 [pF]  (86)

The simulation result of the frequency response with the obtainedconstants are shown in FIG. 16. The amplitude characteristic of theoutput (out0 p) of the 5th-order inverse Chebychev gm-C LPF is identicalto that of FIG. 12. However, the amplitude peak value at the cap02terminal decreases 2.0 times as much by jointing the 1st-order LPF andthe 2nd-order LPF.

Similarly, the frequency response in case where the voltage gain of theamplifier (VCVS) is 10 times as much is shown in FIG. 17. The amplitudeof the output (out0 p) of the 5th-order inverse Chebychev gm-C LPF is10, however, the frequency response remains the same.

The transfer function H(s) of the 5th-order inverse Chebychev LPF isgiven by the equation (39). The derivation method will be furtherdescribed. As in case where the transfer function of the 3rd-orderinverse Chebychev LPF is derived, the transfer function of a 5th-orderChebychev LPF is derived first.

FIG. 18 is a circuit diagram of a 5th-order Chebychev LC ladder LPF. Thetransfer function T_(C)(s) of the 5th-order Chebychev LC ladder LPF isgiven as follows when R1=R3=1, C1=C5=c₁, C₃=c₂, L2=L4=1:

$\begin{matrix}{{T_{c}(s)} = \frac{2}{\begin{matrix}{{c_{1}^{2}l^{2}c_{2}s^{5}} + {2c_{1}l^{2}c_{2}s^{4}} + {\left( {{2c_{1}^{2}l} + {2c_{1}l\; c_{2}} + {l^{2}c_{2}}} \right)s^{3}} +} \\{{\left( {{4c_{1}l} + {2l\; c_{2}}} \right)s^{2}} + {\left( {{2c_{1}} + {2l} + c_{2}} \right)s} + 2}\end{matrix}}} & (87)\end{matrix}$Note that the numerator is set to 2 so that T_(C)(0)=1.

The square of the amplitude characteristic of the equation (87) isgiven:

$\begin{matrix}{{{T_{c}({j\omega})}}^{2} = \frac{4}{\begin{matrix}{\left\{ {2 - {\left( {{4c_{1}l} + {2{lc}_{2}}} \right)\omega^{2}} + {2c_{1}l^{2}c_{2}\omega^{4}}} \right\}^{2} + \left\{ {{\left( {{2c_{1}} + {2l} + c_{2}} \right)\omega} -^{2}} \right.} \\\left. {{\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} + {l^{2}c_{2}}} \right)\omega^{3}} + {c_{1}^{2}l^{2}c_{2}\omega^{5}}} \right\}^{2}\end{matrix}}} & (88)\end{matrix}$

The replacement of the passband by the stopband gives:

$\begin{matrix}{{1 - {{T_{c}({j\omega})}}^{2}} = \frac{\left\{ {{\left( {{2c_{1}} - {2l} + c_{2}} \right)\omega} - {\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} - {l^{2}c_{2}}} \right)\omega^{3}} + {c_{1}^{2}l^{2}c_{2}\omega^{5}}} \right\}^{2}}{\begin{matrix}{\left\{ {2 - {\left( {{4c_{1}l} + {2{lc}_{2}}} \right)\omega^{2}} + {2c_{1}l^{2}c_{2}\omega^{4}}} \right\}^{2} + \left\{ {{\left( {{2c_{1}} + {2l} + c_{2}} \right)\omega} -} \right.} \\\left. {{\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} + {l^{2}c_{2}}} \right)\omega^{3}} + {c_{1}^{2}l^{2}c_{2}\omega^{5}}} \right\}^{2}\end{matrix}}} & (89)\end{matrix}$

The exchange of frequency axes gives:

$\begin{matrix}{{1 - {{T_{c}\left( {j/\omega} \right)}}^{2}} = \frac{\left\{ {{\left( {{2c_{1}} - {2l} + c_{2}} \right)\omega^{4}} - {\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} - {l^{2}c_{2}}} \right)\omega^{2}} + {c_{1}^{2}l^{2}c_{2}}} \right\}^{2}}{\begin{matrix}{\left\{ {{2\omega^{5}} - {\left( {{4c_{1}l} + {2{lc}_{2}}} \right)\omega^{3}} + {2c_{1}l^{2}c_{2}\omega}} \right\}^{2} + \left\{ {{\left( {{2c_{1}} + {2l} + c_{2}} \right)\omega^{4}} -} \right.} \\\left. {{\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} + {l^{2}c_{2}}} \right)\omega^{2}} + {c_{1}^{2}l^{2}c_{2}}} \right\}^{2}\end{matrix}}} & (90)\end{matrix}$

The transfer function H(s) of the 5th-order inverse Chebychev LPF isgiven as follows:

$\begin{matrix}{{H(s)} = \frac{{\left( {{2c_{1}} - {2l} + c_{2}} \right)s^{4}} + {\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} - {l^{2}c_{2}}} \right)s^{2}} + {c_{1}^{2}l^{2}c_{2}}}{\begin{matrix}{{2s^{5}} + {\left( {{2c_{1}} + {2l} + c_{2}} \right)s^{4}} + {\left( {{4c_{1}l} + {2{lc}_{2}}} \right)s^{3}} +} \\{{\left( {{2c_{1}^{2}l} + {2c_{1}{lc}_{2}} + {l^{2}c_{2}}} \right)s^{2}} + {2c_{1}l^{2}c_{2}s} + {c_{1}^{2}l^{2}c_{2}}}\end{matrix}}} & (91)\end{matrix}$

Note that the equation (87) is able to be applied to a 5th-orderButterworth LPF as well. The relationship between the numerator anddenominator in the equation (91) is notable. The second term of thedenominator in the equation (91):{(2_(c) ₁ +2l+_(c) ₂ )ω⁴−(2_(c) ₁ ²l+2_(c) ₁ l_(c) ₂ +l² _(c) ₂ )ω²+_(c)₁ ²l² _(c) ₂ }²and the numerator:{(2_(c) ₁ −2l+_(c) ₂ )ω⁴−(2_(c) ₁ ²l+2_(c) ₁ l_(c) ₂ −l² _(c) ₂ )ω²+_(c)₁ ²l² _(c) ₂ }²can be expressed by:{(2_(c) ₁ ∓2l+_(c) ₂ )ω⁴−(2_(c) ₁ ²l+2_(c) ₁ l_(c) ₂ ∓l² _(c) ₂ )ω² +_(c) ₁ ²l² _(c) ₂ }²Looking at each coefficient of ω^(n), since the numerator has minussigns and the denominator has plus signs, each coefficient of ω^(n) ofthe numerator is smaller than each coefficient of ω^(n) of thedenominator.

By square-rooting the equation (91), the amplitude characteristic of the5th-order inverse Chebychev LPF given by an equation (92) can beobtained. For instance, FIG. 19 shows a graph made by substituting theconstants R1=R3=1 Ω, C1=C5=c₁=0.9766F, C3=c₂=2.0366F, andL2=L4=l=1.6849H of the original 0.01 dB ripple 5th-order Chebychev LPF.

(0.01 dB ripple 5th-order Chebychev LPF:

${{T_{c}(s)} = \frac{2}{\begin{matrix}{{5.514263\mspace{11mu} s^{\; 5}} + {11.29278\mspace{11mu} s^{\; 4}} + {15.69796\mspace{11mu} s^{\; 3}} +} \\{{13.44483\mspace{11mu} s^{\; 2}} + {7.3596\mspace{11mu} s} + 2}\end{matrix}}};$5th-order Butterworth LPF:

${{T(s)} = \frac{2}{\begin{matrix}{{{2\; s^{\; 5}}\; + \;{6.47135955\mspace{11mu} s^{\; 4}}\; + \;{10.47213595\mspace{11mu} s^{\; 3}}\; +}\;} \\{{10.47213595\mspace{11mu} s^{\; 2}}\; + \;{6.47135955\mspace{11mu} s}\; + \; 2}\end{matrix}}};$5th-order inverse cbc LPF:

$\left. {{H(s)} = \frac{{0.62\; s^{4}} + {4.134601\; s^{2}} + 5.514263}{\begin{matrix}{{2s^{5}} + {7.3596\mspace{11mu} s^{4}} + {13.44483\mspace{11mu} s^{3}} +} \\{{15.69796\mspace{11mu} s^{\; 2}} + {11.29278\mspace{11mu} s} + 5.514263}\end{matrix}}} \right)$

Although the amplitude characteristic of the 0.01 dB ripple 5th-orderChebychev LPF and the amplitude characteristic of the 5th-orderButterworth LPF are almost the same in the passband, marked differencesin the attenuation characteristic are seen in the stopband. Theequiripple characteristic of the stopband does not attenuate as much as30 dB in the amplitude characteristic of the inverse Chebychev LPF.Further, the amplitude characteristic of the inverse Chebychev LPF isbetter than the attenuation characteristic of the 0.01 dB ripple5th-order Chebychev LPF until the first transmission zero, however,after that, the attenuation characteristic of the inverse Chebychev LPFis not as much as that of the 0.01 dB ripple 5th-order Chebychev LPF.

However, the amplitude characteristic of the inverse Chebychev LPF issuperior to that of the 5th-order Butterworth LPF until the point justbeyond the second transmission zero.

FIG. 20 shows the phase characteristic of the 5th-order inverseChebychev LPF along with the phase characteristics of the 0.01 dB ripple5th-order Chebychev LPF and the 5th-order Butterworth LPF.

The phase characteristic of the 5th-order inverse Chebychev LPF issmoother than, and superior to the phase characteristics of the 0.01 dBripple 5th-order Chebychev LPF and the 5th-order Butterworth LPF in thepassband (ω<1).

Now we will return to the description of how the transfer function ofthe 5th-order Chebychev LPF is factorized.

The transfer function of the 0.01 dB ripple 5th-order Chebychev LPF isgiven as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{0.362696}{\begin{matrix}{\left( {s + 0.6328} \right)\left( {s^{\; 2} + {1.024\mspace{11mu} s} + 0.60777} \right)} \\\left( {s^{\; 2} + {0.3912\mspace{11mu} s} + 0.94304} \right)\end{matrix}}} & (92)\end{matrix}$

Expanding the denominator in the expression (92) gives:

$\begin{matrix}{{T_{c}(s)} = \frac{0.362696}{\begin{matrix}{s^{5} + {2.047921\mspace{11mu} s^{4}} + {2.846792\mspace{11mu} s^{3}} +} \\{{2.438191\mspace{11mu} s^{2}} + {1.334648\mspace{11mu} s} + 0.3962696}\end{matrix}}} & (93)\end{matrix}$

The equation (93) is the equation (86) with the denominator and thenumerator divided by c1212c2=5.514263, and the same transfer function isgiven.

The equation (92) is derived from the pole locations of the 5th-orderChebychev LPF. The equations (8) through (14) are similarly applied withn=5(5th-order) in the derivation method.

For instance, in order to obtain an equiripple attenuationcharacteristic of −40 dB with an inverse Chebychev LPF,α_(max)=0.0004342 dB and ε=0.01. From the equation (16), a=1.059685274and sin h(a)=1.269449. Further, Φ_(K)=18°, 54°, and 0°, and thefollowing is derived:p ₁ ,p ₂=−0.392281314±j1.536920645p ₃ ,p ₄=−1.027005815±j0.949869196p ₅=−1.269449

The transfer function of the 0.0004342 dB ripple 5th-order Chebychev LPFis given as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{6.250525}{\begin{matrix}{\left( {s + 1.2694489} \right)\left( {s^{2} + {0.784563\mspace{11mu} s} + 2.516009} \right)} \\\left( {s^{2} + {2.054012\mspace{11mu} s} + 1.956992} \right)\end{matrix}}} & (94)\end{matrix}$

Expanding the denominator in the expression (94) gives:

$\begin{matrix}{{T_{c}(s)} = \frac{6.250525}{\begin{matrix}{s^{5} + {4.108023\; s^{4}} + {9.687927\mspace{11mu} s^{3}} +} \\{{14.42726\mspace{11mu} s^{2}} + {13.4333\mspace{11mu} s} + 6.250525}\end{matrix}}} & (95)\end{matrix}$

Note that the cut-off frequency ω_(hp) (hp:half-power) is similarlygiven by the equation (17), and in the case of the 0.0004342 dB ripple5th-order Chebychev LPF, the cut-off frequency ω_(hp)=1.616001. When thecut-off frequency ω_(hp)=1, the transfer function of the 0.0004342 dBripple 5th-order Chebychev LPF is given as follows:

$\begin{matrix}{{T_{c}(s)} = \frac{0.567163}{\begin{matrix}{s^{5} + {2.542092\mspace{11mu} s^{4}} + {3.709775\mspace{11mu} s^{3}} +} \\{{3.418683\mspace{11mu} s^{2}} + {1.969773\mspace{11mu} s} + 0.567163}\end{matrix}}} & (96)\end{matrix}$

When the denominator is factorized, the transfer function is given asfollows:

$\begin{matrix}{{T_{c}(s)} = \frac{0.567163}{\begin{matrix}{\left( {s + 0.78555} \right)\left( {s^{2} + {0.485496\mspace{11mu} s} + 0.963449465} \right)} \\\left( {s^{2} + {1.271046\mspace{11mu} s} + 0.749386387} \right)\end{matrix}}} & (97)\end{matrix}$

Comparing the coefficients of the equations (87) and (96),c₁=0.786753587, l=1.696112321, and c₂=1.980320286 hold. Note that c₁, l,and c₂ must be selected so that the 5^(th) through 0^(th)-ordercoefficients are identical. The values of c₁, l, and c₂ with which the5^(th), 4th and 0th-order coefficients are the same but the 3rd and2nd-order coefficients differ by a few percents exist.

When these values are substituted for the equation (78), the transferfunction H(s) of the 5th-order inverse Chebychev LPF is given by thefollowing equation:

$\begin{matrix}{{H(s)} = \frac{{0.161603\mspace{11mu} s^{4}} + {1.687912\mspace{11mu} s^{2}} + 3.526323}{\begin{matrix}{{2s^{5}} + {6.946052\mspace{11mu} s^{4}} + {12.05538\mspace{11mu} s^{3}} +} \\{{13.08187\mspace{11mu} s^{2}} + {8.964238\mspace{11mu} s} + 3.526323}\end{matrix}}} & (98)\end{matrix}$Or:

$\begin{matrix}{{H(s)} = \frac{{0.045828\mspace{11mu} s^{4}} + {0.47866\mspace{11mu} s^{2}} + 1}{\begin{matrix}{{0.567163\mspace{11mu} s^{5}} + {1.969772\mspace{11mu} s^{4}} + {{.3418683}\mspace{11mu} s^{3}} +} \\{{3.709776\mspace{11mu} s^{2}} + {2.542092\mspace{11mu} s} + 1}\end{matrix}}} & (99)\end{matrix}$

From the equation (96), the denominator of the transfer function H(s) ofthe 5th-order inverse Chebychev LPF is as follows:

$\begin{matrix}{{H(s)} = \frac{{as}^{4} + {bs}^{2} + 1}{\begin{matrix}{{0.567163\mspace{11mu} s^{\; 5}} + {1.969773\mspace{11mu} s^{\; 4}} + {3.418683\mspace{11mu} s^{\; 3}} +} \\{{3.709775\mspace{11mu} s^{\; 2}} + {2.542092\mspace{11mu} s} + 1}\end{matrix}}} & (100)\end{matrix}$

The denominator of the equation (100) is the same as that of theequation (99). The numerator of the equation (100) is the same as thatof the equation (99) as well. Therefore, a=0.045828 and b=0.47866 in thenumerator of the equation (100). Similarly, from the equation (97), thedenominator of the transfer function H(s) of the 5th-order inverseChebychev LPF is factorized as follows:

$\begin{matrix}{{H(s)} = \frac{{0.0458276\mspace{11mu} s^{4}} + {0.47866\mspace{11mu} s^{2}} + 1}{\begin{matrix}{\left( {1 + {0.78555\mspace{11mu} s}} \right)\left( {1 + {0.485496\mspace{11mu} s} + {0.963449465\mspace{11mu} s^{2}}} \right)} \\\left( {1 + {1.271046\mspace{11mu} s} + {0.749386387\mspace{11mu} s^{2}}} \right)\end{matrix}}} & (101)\end{matrix}$

Therefore, the transfer function H(s) of the 5th-order inverse ChebychevLPF is given as follows:

$\begin{matrix}{{H(s)} = \frac{\left( {{0.3463405\mspace{11mu} s^{2}} + 1} \right)\left( {{0.13231949\mspace{11mu} s^{2}} + 1} \right)}{\begin{matrix}{\left( {1 + {0.78555\mspace{11mu} s}} \right)\left( {1 + {0.485496\mspace{11mu} s} + {0.963449465\mspace{11mu} s^{2}}} \right)} \\\left( {1 + {1.271046\mspace{11mu} s} + {0.749386387\mspace{11mu} s^{2}}} \right)\end{matrix}}} & (102)\end{matrix}$

The amplitude characteristic of the 5th-order Chebychev LPF obtained asdescribed above is shown in FIG. 9 (characteristic c in FIG. 9), and thephase characteristic is shown in FIG. 10 (characteristic a in FIG. 10.)

How to derive transfer functions of 3rd-order and 5th-order inverseChebychev LPFs have been described above. Further, in the examples ofinverse Chebychev LPFs, only ones with a ripple of −18 dB or around −20dB in the attenuation band have been shown. This may give an impressionthat much attenuation cannot be obtained.

However, as shown in examples of −26 dB and −40 dB, a desired equiripplecharacteristic in the attenuation band can be obtained. Regardinginverse Chebychev LPFs, even-order LPFs such as 2nd-order and 4th-orderLPFs and 7th or higher odd-order LPFs are defined, however, as one caneasily conjecture from the method for deriving the 5th-order transferfunction, the method for deriving high-order transfer functions comewith a great deal of difficulty.

However, the following generalization can be made from theabove-described methods for deriving the transfer functions of theinverse Chebychev LPFs.

α_(min)[dB] is the equiripple characteristic in the attenuation band.Now, when α_(mi) is set to 60 dB and n=7 (7th-order), ε=0.001(=10−3),therefore 20 log(ε)=−α_(min), and the following equation is given:

$\begin{matrix}{ɛ = {10^{- \frac{\alpha_{\min}}{20}} = \sqrt{10^{- \frac{\alpha_{\min}}{10}}}}} & (103)\end{matrix}$

Next, equation (8) is used as it is:

$\begin{matrix}{a = {\frac{1}{n}{\sinh^{- 1}\left( \frac{1}{ɛ} \right)}}} & (104)\end{matrix}$

Then, k=0, 1, 2, 3 are substituted using the following:

$\begin{matrix}{{\phi_{k} = {{\frac{{2k} + 1}{n}\frac{\pi}{2}\mspace{31mu} k} = 0}},1,\ldots\mspace{11mu},{{2n} - 1}} & (105)\end{matrix}$As a result, 12.8571°, 38.5714°, 64.2857°, and 90° are obtained.

Now the following equations are calculated:−α_(K)=sin h(a)·sin(Φ_(K))  (106)±β_(K)=cos h(a)·cos(Φ_(K))  (107)

Since cos(90°)=0, the followings hold:a ₀=α₃/cos h(a)  (108)b _(k)=2α_(K)/cos h(a)  (109)c _(K)=(α_(K) ²+β_(K) ²)/cos h ²(a)  (110)

Further, as for the coefficients of the denominators, the followinghold:Ω₂=sin(π/n)/cos h ²(a),Ω₄=sin(2π/n)/cos h ²(a),Ω₆=sin(4π/n)/cos h ²(a)  (111)

Based on the above, the following equations are given, for an odd-order(n=3, 5, 7, . . . ):

$\begin{matrix}{\mspace{85mu}{{H(s)} = \frac{\prod\limits_{=}^{{({n -})}/2}{\left( {1 + {\Omega_{2i}^{2}s^{2}}} \right)}}{\left( {1 + {a_{0}s}} \right){\prod\limits_{=}^{{({n -})}/2}\left( {1 + {b_{2i}s} + {c_{2i}s^{2}}} \right)}}}} & (112)\end{matrix}$and for an even-order (n=2, 4, 6, . . . ):

$\begin{matrix}{\mspace{79mu}{{H(s)} = \frac{\prod\limits_{=}^{n/2}\left( {1 + {\Omega_{2i}^{2}s^{2}}} \right)}{\prod\limits_{=}^{n/2}\left( {1 + {b_{2i}s} + {c_{2i}s^{2}}} \right)}}} & (113)\end{matrix}$

The present invention has been described in detail using inverseChebychev filters with elliptic characteristics as examples. The methodof embodying the present invention to the inverse Chebychev filterdescribed in detail, can be similarly applied to elliptic filters (cauerfilters). The method for deriving the transfer function of an ellipticfilter (cauer filter) is described in routine textbooks. Further, asmentioned above, the methods for deriving the transfer functions of aButterworth filter, Chebychev filter, and Bessel filter are similarlydescribed in routine text books, and a normalized constant of eachelement in the LCR ladder filter is usually found in data books andwidely known. So detailed description thereof will be omitted here.

However, such a Butterworth filter, Chebychev filter, and Bessel filterare realized by eliminating the elliptic capacitors C2 and C4 from thegm-C filter circuits shown in FIGS. 3, 11, and 15, respectively and thetransfer function can be obtained by making the elliptic capacitors C2and C4 zero.

For instance, if the elliptic capacitors C2 is eliminated from the2nd-order elliptic gm-C LPF in the succeeding stage of the amplifier ofthe 3rd-order inverse Chebychev gm-C LPF shown in FIG. 3, it will becomea 2nd-order gm-C LPF shown in FIG. 21, and when C2=0 in the equation(23), the transfer function of the 2nd-order gm-C LPF is obtained. Thetransfer function T2(s) is expressed as follows:

⁢T 2 ⁡ ( s ) = V out V in = 4 ⁢ g m ⁢ ⁢ 3 ⁢ g m ⁢ ⁢ 4 C ⁢ L2 ⁢ C 3 s 2 + s ⁢ 2 ⁢ g m⁢⁢6 C 3 + 4 ⁢ g m ⁢ ⁢ 4 ⁢ g m ⁢ ⁢ 6 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 ( 114 )

Note that, in the equation (114), since g_(m4), g_(m5), and g_(m6) arecoefficients in the denominator, they determine the frequency response.On the other hand, since g_(m3) is in the numerator only, it does notinfluence the frequency response, but determines the voltage gain of the2nd-order LPF. In other words, the voltage gain of the 2nd-order LPF canbe varied by increasing or decreasing the g_(m) value of the OTA in thefirst input-stage.

However, as described in detail in the present application, in generalit is set up so that g_(m3)=g_(m4)=g_(m5)=g_(m6)=g_(m), and the equation(114) is expressed as follows:

⁢T 2 ⁡ ( s ) = V out V in = 4 ⁢ g m 2 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 s 2 + s ⁢ 2 ⁢ g m C 3 + 4⁢g m 2 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 ( 115 )

It is not usually understood that the transconductance gm of the OTA ofthe 2nd-order LPF in the first input-stage is a constant determining thevoltage gain of the 2nd-order LPF and that it does not influence thefrequency response.

In other words, the amplifying means described in the presentapplication is equivalently realized by increasing or decreasing the gmof the OTA in the first input-stage of the 2nd-order LPF whilemaintaining the capacitance value and the frequency response.

This method is a very effective technique for correcting an insertionloss of the 2nd-order filter. As concrete examples, there are techniquessuch as varying the value of g_(m) of the OTA by varying the drivecurrent of the OTA in the first input-stage of the 2nd-order LPF, ormultiplying the voltage gain approximately by an integer i.e.approximately multiplying the g_(m) by an integer by varying the numberof the OTA connected in parallel.

Embodiment 4

From the above descriptions, it is apparent that even when usingamplifying means employing a variable gain amplifier, of which the gaincan be varied instead of an amplifier with a fixed voltage gain, thefrequency response is not influenced and only voltage gain is varied. Inother words, as shown in FIG. 22, the voltage gain may be varied bymaking amplifiers 1 and 2 variable gain amplifiers.

Embodiment 5

With the Butterworth filter, Chebychev filter, and Bessel filter, whichdon't need any elliptic capacitor, as a 5th-order gm-C LPF shown in FIG.23, the function of the variable gain amplifiers (amplifiers 1 and 2)can be equivalently achieved by varying the g_(m) values of OTAs in thefirst input-stages of (a 1st-order gm-C LPF and) two 2nd-order gm-CLPFs.

The transfer function T(s) of the 5th-order gm-C LPF is expressed asfollows:

T ⁡ ( s ) = T 1 ⁡ ( s ) × T 2 ⁡ ( s ) × T 3 ⁡ ( s ) = 2 ⁢ g m ⁢ ⁢ 1 C 1 s + 2 ⁢g m ⁢ ⁢ 2 C 1 × 4 ⁢ g m ⁢ ⁢ 3 ⁢ g m ⁢ ⁢ 4 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 s 2 + 2 ⁢ g m ⁢ ⁢ 6 C 3 ⁢s + 4 ⁢ g m ⁢ ⁢ 4 ⁢ g m ⁢ ⁢ 6 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 × 4 ⁢ g m ⁢ ⁢ 7 ⁢ g m ⁢ ⁢ 8 C ⁢ L ⁢ ⁢ 2 ⁢ C3 s 2 + 2 ⁢ g m ⁢ ⁢ 10 C 3 ⁢ s + 4 ⁢ g m ⁢ ⁢ 8 ⁢ g m ⁢ ⁢ 9 C ⁢ L ⁢ ⁢ 2 ⁢ C 3 ( 116 )

For instance, the transfer function T(s) in the case of a 5th-orderButterworth LPF is expressed and factorized as follows:

$\begin{matrix}\begin{matrix}{{T(s)} = \frac{1}{s^{5} + {3.236{.68}\mspace{11mu} s^{4}} + {5.23607\mspace{11mu} s^{3}} + {5.23607\mspace{11mu} s^{2}} + {3.236{.68}\mspace{11mu} s} + 1}} \\{= \frac{1}{\left( {s + 1} \right)\left( {s^{2} + {1.603034\mspace{11mu} s} + 1} \right)\left( {s^{2} + {0.618034\mspace{11mu} s} + 1} \right)}}\end{matrix} & (117) \\{{T_{1}(s)} = \frac{1}{s + 1}} & (118) \\{{T_{1}(s)} = \frac{1}{s^{2} + {1.603034\mspace{11mu} s} + 1}} & (119) \\{{T_{1}(s)} = \frac{1}{s^{2} + {0.618034\mspace{11mu} s} + 1}} & (120)\end{matrix}$

Here, in the 5th-order gm-C LPF shown in FIG. 23, when respectivelymultiplying each g_(m) value of an OTA1, OTA3, and OTA7 by G_(V0),G_(V1), and G_(V2), thus making the values G_(V0)g_(m1), G_(V1)g_(m3),and G_(V2)g_(m7), the transfer functions T(s) is given as follows:

$\begin{matrix}{\mspace{65mu}\begin{matrix}{{T(s)} = {{T_{1}(s)} \times {T_{2}(s)} \times {T_{3}(s)}}} \\{G_{F\; 0} \times G_{F\; 1} \times G_{F\; 2} \times \frac{\frac{2g_{m\; 1}}{C_{1}}}{s + \frac{2g_{m\; 2}}{C_{1}}} \times} \\{\frac{\frac{4g_{m\; 3}g_{m\; 4}}{C_{L2}C_{3}}}{s + {\frac{2g_{m\; 6}}{C_{3}}s} + \frac{4g_{m\; 4}g_{m\; 6}}{C_{L2}C_{3}}} \times \frac{\frac{4g_{m\; 7}g_{m\; 8}}{C_{L2}C_{3}}}{s^{2} + {\frac{2g_{m\; 10}}{C_{3}}s} + \frac{4g_{m\; 8}g_{m\; 9}}{C_{L2}C_{3}}}}\end{matrix}} & (121)\end{matrix}$

For instance, the transfer function T(s) in case of the 5th-orderButterworth LPF is expressed as follows:

$\begin{matrix}\begin{matrix}{{T(s)} = \frac{G_{F\; 0} \times G_{F\; 1} \times G_{F\; 2}}{s^{5} + {3.236068\mspace{11mu} s^{4}} + {5.23607\mspace{11mu} s^{3}} + {5.23607\mspace{11mu} s^{2}} + {3.236068\mspace{11mu} s} + 1}} \\{= \frac{G_{F\; 0} \times G_{F\; 1} \times G_{F\; 2}}{\left( {s + 1} \right)\left( {s^{2} + {1.603034\mspace{11mu} s} + 1} \right)\left( {s^{2} + {0.618034\mspace{11mu} s} + 1} \right)}}\end{matrix} & (122)\end{matrix}$and the followings hold:

$\begin{matrix}{{T_{1}(s)} = \frac{G_{F\; 0}}{s + 1}} & (123) \\{{T_{1}(s)} = \frac{G_{F\; 1}}{s^{2} + {1.603034\mspace{11mu} s} + 1}} & (124) \\{{T_{1}(s)} = \frac{G_{F\; 2}}{s^{2} + {0.618034\mspace{11mu} s} + 1}} & (125)\end{matrix}$

When the equations (116) and (121) are compared, one can see that thefrequency response remains the same, but the level is multiplied byG_(V0)×G_(V1)×G_(V2). In other words, an amplifier 0 whose voltage gainis multiplied by G_(V0), an amplifier 1 whose voltage gain is multipliedby G_(V1), and an amplifier 2 whose voltage gain is multiplied by G_(V2)are equivalently obtained. It is understood that variable values can beeasily created by varying the drive current of the OTAs and the g_(m)values instead of setting the values of G_(V0), G_(V1), and G_(V0) tofixed values.

The present invention is applied to a filter circuit integrated on a LSIchip. The present invention is suitably applied to the correction of aninsertion loss of the filter circuit itself caused by the variance inthe fabrication process or of a shift in the voltage gain of the filter,and to a receiving channel selection filter of a chip for a mobilewireless terminal device. When it is applied to the receiving channelselection filter, a plurality of high-voltage gain amplifiers such asVGAs and PGAs whose gains are varied with an exponential characteristicare implemented, and the present application is very effective formaintaining desirable noise characteristics.

It should be noted that other objects, features and aspects of thepresent invention will become apparent in the entire disclosure and thatmodifications may be done without departing the gist and scope of thepresent invention as disclosed herein and claimed as appended herewith.

Also it should be noted that any combination of the disclosed and/orclaimed elements, matters and/or items may fall under the modificationsaforementioned.

1. A filter circuit having an order of three or more and a cutofffrequency f_(c), said filter circuit comprising at least one amplifyingunit for amplifying an in-band signal, a frequency response at an outputof said filter circuit having an attenuation characteristic obtainablewith said order of said filter circuit, based upon an attenuationmeasured at a frequency 2 f _(c) and an expected attenuation of about 6dB/octave per order, comprising a plurality of stages, each comprisingat least one of a 1st-order filter and a 2nd-order filter, which arecascade-connected to each other, a summation of orders of said 1st-orderand 2nd-order filters being (2×m+1), where m is an integer equal to orgreater than 1; wherein said amplifying unit comprises an amplifierarranged between an output of a first-stage filter and an input of asucceeding, second-stage filter, in at least one pair among a pluralityof pairs of filters of adjacent stages.
 2. The filter circuit accordingto claim 1, wherein a gain of said amplifying unit is variablycontrolled.
 3. The filter circuit according to claim 1, comprising a1st-order filter and m number of 2nd-order filter or filters, which arecascade-connected to each other; wherein said amplifying unit comprisesan amplifier arranged between an output of a first-stage filter and aninput of a succeeding, second-stage filter in at least one pair among mpairs of filters of adjacent stages.
 4. The filter circuit according toclaim 1, wherein a noise component amplified by said amplifier isattenuated by the succeeding-stage filter.
 5. The filter circuitaccording to claim 1, wherein at least one of said 1st-order filter andsaid 2nd-order filter comprises a gm-C filter including an OTA(Operational Transconductance Amplifier) and a capacitor which ischarged with a current output from the OTA, said current correspondingto an input voltage supplied to the OTA.
 6. The filter circuit accordingto claim 1, wherein a voltage gain of said amplifier is variablycontrolled.
 7. The filter circuit according to claim 5, wherein saidgm-C filter constituting said 1st-order filter and/or said 2nd-orderfilter includes at least one OTA whose mutual conductance gm is variablycontrolled.
 8. The filter of claim 1, said filter comprising an inversefilter wherein a pass band and a stop band relationship is functionallyinversed in a design process of said filter.
 9. The filter of claim 8,wherein said inverse filter comprises an inverse Chebychev filter. 10.The filter of claim 1, as incorporated on a chip.
 11. A mobile wirelessterminal device, incorporating the chip of claim 10 as a receivingchannel selection filter.
 12. A filter circuit having an order of threeor more and a cutoff frequency f_(c), said filter circuit comprising atleast one amplifying unit for amplifying an in-band signal, a frequencyresponse at an output of said filter circuit having an attenuationcharacteristic obtainable with said order of said filter circuit, basedupon an attenuation measured at a frequency 2 f _(c) and an expectedattenuation of about 6 dB/octave per order, comprising a plurality ofstages, each comprising at least one of a 1st-order filter and/or a2nd-order filter, which are cascade-connected to each other, a summationof the orders of said 1st-order and 2nd-order filters being 2m, where mis an integer equal to or greater than 2, wherein said amplifying unitcomprises an amplifier arranged between an output of a first-stagefilter and an input of a succeeding, second-stage filter in at least onepair among a plurality of pairs of filters of adjacent stages.
 13. Thefilter circuit according to claim 12, wherein a noise componentamplified by said amplifier is attenuated by the succeeding-stagefilter.
 14. The filter circuit according to claim 12, wherein at leastone of said 1st-order filter and said 2nd-order filter comprises a gm-Cfilter including an OTA (Operational Transconductance Amplifier) and acapacitor which is charged with a current output from the OTA, saidcurrent corresponding to an input voltage supplied to the OTA.
 15. Thefilter circuit according to claim 12, wherein a voltage gain of saidamplifier is variably controlled.
 16. The filter circuit according toclaim 14, wherein said gm-C filter constituting said 1st-order filterand/or said 2nd-order filter includes at least one OTA whose mutualconductance gm is variably controlled.
 17. The filter circuit accordingto claim 12, wherein a gain of said amplifying unit is variablycontrolled.
 18. A filter circuit having an order of three or more,comprising: at least one 2^(nd)-order LPF stage; a plurality of low passfilter (LPF) stages cascaded-connected to each, each said LPF stagecomprising one of a 1^(st)-order LPF stage and a 2^(nd)-order LPF stage,at least one of said 1^(st)-order LPF and 2^(nd)-order LPF stagescomprising a gm-C filter including an OTA (Operational TransconductanceAmplifier) and a capacitor which is charged with a current output fromthe OTA, said current corresponding to an input voltage supplied to theOTA; and an amplifier interconnecting said at least one 2^(nd)-order LPFstage with said cascaded-connected LPF stages, wherein an amplifierinterconnects each said LPF stage in said plurality of stages.
 19. Thefilter of claim 18, wherein said filter comprises one of: an inverseChebychev filter wherein a pass band and a stop band relationship isfunctionally inversed in a design process; an elliptic (cauer) filter; aButterworth filter; a Chebychev filter; and a Bessel filter.
 20. Afilter circuit having an order of three or more, comprising: a pluralityof low pass filter (LPF) stages cascaded-connected to each other, eachsaid stage comprising one of a 1^(st)-order LPF stage and a 2^(nd)-orderLPF stage, at least one of said 1^(st)-order LPF and 2^(nd)-order LPFstages comprising a gm-C filter including an OTA (OperationalTransconductance Amplifier) and a capacitor which is charged with acurrent output from the OTA, said current corresponding to an inputvoltage supplied to the OTA; and an amplifier interconnecting each saidLPF stage in said plurality of stages, wherein: each said 1^(st)-orderLPF stage comprises: a first OTA having a differential input and adifferential output; a second OTA having a differential input and adifferential output, said differential input of said second OTA beingconnected to said differential output of said first OTA, saiddifferential output of said second OTA being connected to saiddifferential input of said second OTA; and first and second capacitors,first connections of said first and second capacitors being connected toa ground and second connections of said first and second capacitorsbeing connected to said differential output of said second OTA; and eachsaid 2^(nd)-order LPF stage comprises: a third OTA having a differentialinput and a differential output and having associated therewith thirdand fourth capacitors, first connections of said third and fourthcapacitors being connected to said ground and second connections of saidthird and fourth capacitors being connected to said differential outputof said OTA; a fourth OTA having a differential input and a differentialoutput, said differential input of said fourth OTA being connected tosaid differential output of said third OTA; a fifth OTA having adifferential input and a differential output, said differential input ofsaid fifth OTA being connected to said output of said fourth OTA, saiddifferential output of said fifth OTA being connected to saiddifferential input of said fourth OTA in a crisscross manner; a sixthOTA having a differential input and a differential output, saiddifferential input of said sixth OTA being connected to saiddifferential input of said sixth OTA and to said differential input ofsaid fifth OTA; fifth and sixth capacitors, first connections of saidfifth and sixth capacitors being connected to said differential input ofsaid third OTA and second connections of said fifth and sixth capacitorsbeing connected to said differential output of said fourth OTA; andseventh and eight capacitors, first connections of said seventh andeight capacitors being connected to said ground and second connectionsof said seventh and eight capacitors being connected to saiddifferential output of said fourth OTA.